《線性代數(shù)應該這樣學》課內驗證2-有限維向量空間

CHAPTER 2 Finite-Dimensional Vector Spaces

  1. If some vectors are removed from a linearly independent list, the remaining
    list is also linearly independent, as you should verify.

Proof Suppose v_1,\cdots ,v_m, \cdots ,v_n is linearly independent, then the only choice of a_1, \cdots, a_n\in \mathbb{F} that makes a_1 v_1 +\cdots +a_n v_n=0 is a_1=\cdots = a_n=0.

Since the order of vectors does not affect linear independence, we put all the vectors that need to be removed on the right.

Now assume the vectors v_{m+1}, \cdots ,v_n are removed, and the remaining v_1, \cdots, v_m is linearly dependent.

Hence there exist a_1,\cdots ,a_m \in \mathbb{F}, not all 0, such that a_1 v_1 +\cdots + a_m v_m=0.

Let a_{m+1},\cdots,a_n all are 0, hence a_1 v_1+\cdots+a_m v_m+\cdots+a_nv_n=0, at the same time a_1,\cdots,a_n, not all 0. We get a contradiction.

Thus the remaining lise is also linearly independent.

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